[Part I] The History of Subitizing: From Early Psychophysical Experiments to Modern Theories of Visual Attention

Abstract

The ability to rapidly determine the number of objects is one of the fundamental characteristics of human visual perception. Unlike sequential counting, which requires conscious effort and time, small sets of objects are perceived almost instantaneously. Research on this phenomenon first led to the discovery of subitizing - a mechanism for the rapid and accurate identification of small numerosities - and later, in the modern era, to the development of the Approximate Number System (ANS), which explains the approximate perception of larger quantities. Over the past seven decades, theories of numerical perception have evolved considerably: from the idea of exceptionally rapid counting to contemporary models integrating visual attention, working memory, visual segmentation, and neural representations of numerical magnitude.

Introduction: How We Perceive Small Quantities

The ability to estimate the number of objects without using language or formal arithmetic long remained largely unnoticed. In everyday life, people effortlessly recognize one, two, or three objects without consciously reflecting on how this judgment is made. However, once the number of objects increases to five, six, or seven, the processing strategy changes markedly: sequential counting or approximate estimation becomes necessary.

Although this distinction may appear obvious, it became the starting point for one of the longest-running research programs in cognitive psychology. Over the past several decades, researchers have sought to answer a number of fundamental questions. Why are small numerosities perceived almost instantaneously? Does the brain employ a specialized mechanism for processing small sets, or is subitizing simply an extremely rapid form of counting? How are mechanisms for the precise identification of small quantities related to those responsible for the approximate estimation of larger sets? Do specialized neural systems exist for processing numerical information?

The search for answers to these questions gradually led to modern theories of numerical cognition. However, the path toward these theories proved to be considerably more complex than initially expected.

Early Studies: The Discovery of an Unusual Phenomenon

The modern study of numerical perception is generally traced back to the landmark work of Kaufman, Lord, Reese, and Volkmann, published in 1949. Their objective was straightforward: to measure how long it takes a person to determine the number of simultaneously presented objects.

Participants were briefly shown sets of dots containing different numbers of elements and were asked to report the number of dots they had seen. Intuitively, one would expect reaction time to increase gradually with every additional object - the more elements that must be counted, the longer the response should take.

The experimental results, however, revealed a far more intriguing pattern. For small sets, reaction time remained almost independent of the number of objects presented.

For example:

Number of Objects	Reaction Time
1	≈410 ms
2	≈430 ms
3	≈450 ms

Beginning at approximately four objects, however, the relationship changed dramatically. Each additional object increased reaction time by roughly 250-350 milliseconds.

Although later studies demonstrated that neither region is perfectly linear - the subitizing region exhibits a slight positive slope, while the enumeration region is only approximately linear - Kaufman and colleagues were the first to demonstrate the existence of two qualitatively different modes of numerical processing. Although they did not propose a definitive explanation for their findings, their work laid the foundation for the study of the phenomenon that would later become known as subitizing.

Why Does This Transition Occur?

Early interpretations were relatively simple. It was assumed that people always relied on sequential counting, but that for small numbers of objects this counting occurred so quickly that the process appeared instantaneous. However, numerous experiments were difficult to reconcile with this explanation.

First, reaction times for one, two, and three objects differed only slightly. Second, errors in judging small numerosities were extremely rare. This suggested the existence of a qualitatively different mechanism.

One of the most influential attempts to explain the nature of subitizing was the review by Trick and Pylyshyn (1994). After analyzing a wide range of experimental findings, the authors concluded that subitizing is not a form of rapid counting. In their view, numerical processing begins much earlier. Before determining the number of objects, the visual system must answer a more fundamental question:

Which objects are present in the visual scene?

To describe this process, Trick and Pylyshyn introduced the concept of object files. According to this model, each detected object receives its own temporary representation - an object file - containing information about its position, color, shape, and other features. The number of such simultaneously maintained representations is limited, typically to about three or four objects. If only a few objects are present, all of them can be represented almost simultaneously, allowing their quantity to be identified quickly and accurately.

If the number of objects increases, attention must begin to shift from one object to another, which leads to a gradual increase in reaction time. Thus, Trick and Pylyshyn proposed that subitizing should be understood not as an arithmetic process, but as a consequence of the limited capacity of the visual system to individualize several independent objects at the same time. This theory significantly changed the direction of research. After its emergence, the focus shifted from the study of counting itself to the mechanisms of attention, visual perception, and object individuation.

Conclusion

Research on subitizing has progressed from early psychophysical experiments to modern models of visual attention. The introduction of the concept of object files made it possible to view subitizing as a manifestation of the limited capacity of the visual system to simultaneously individuate several independent objects, rather than as a special case of arithmetic processing. This idea had a major influence on subsequent research in attention, visual perception, and working memory.

Despite this progress, the object-file theory did not answer another fundamental question. If subitizing mechanisms allow people to identify small sets quickly and accurately, how are they able to estimate much larger quantities just as rapidly, when the number of objects greatly exceeds the limit of simultaneous individuation? In such cases, sequential counting is clearly too slow, and the object-file model can no longer fully explain the observed behavior.

The search for an answer to this question became one of the reasons for the emergence of a new research direction devoted to the approximate perception of quantity. Gradually, researchers shifted their attention from mechanisms of precise identification of small sets to processes of rapid estimation of large numerosities, eventually leading to the development of the concept of the Approximate Number System. The origins of this theory, its experimental foundations, and contemporary views on the nature of approximate numerical perception will be discussed in the second part of this review.